Tensor product of super-vector spaces

Question

We have an isomorphism $\mathbb{C}^m\otimes\mathbb{C}^n\cong\mathbb{C}^{mn}$.

But what happens if we consider super-vector spaces $$\mathbb{C}^{p|q}\otimes\mathbb{C}^{r|s}?$$

To what is this isomoprhic? Maybe to $\mathbb{C}^{(p+q)(r+s)|0}$?

Answer 1

\begin{align}(\mathbb{C}^{p|q}\otimes\mathbb{C}^{r|s})_0&=(\mathbb{C}^{p}\otimes\mathbb{C}^{r})\oplus(\mathbb{C}^{q}\otimes\mathbb{C}^{s})\\&=\mathbb{C}^{pr}\oplus \mathbb{C}^{qs}\\&=\mathbb{C}^{pr+qs}\\\end{align}

A similar argument for the degree $1$ shows that \begin{align}\mathbb{C}^{p|q}\otimes\mathbb{C}^{r|s}&\simeq\mathbb{C}^{pr+qs| ps+qr}\\\end{align}

Answer 2

No. It's isomorphic to $\mathbb{C}^{pr+qs|ps+qr}$. That's because, given two super vector spaces $V$ and $W$,$$(V\otimes W)_{\overline 0}=V_{\overline 0}\otimes W_{\overline 0}\oplus V_{\overline 1}\otimes W_{\overline 1}$$and$$(V\otimes W)_{\overline 1}=V_{\overline 0}\otimes W_{\overline 1}\oplus V_{\overline 1}\otimes W_{\overline 0}.$$